You can use preserve for that: add to you type-definition (e.g. THREE_LEVEL) the line
preserve[LevelThree+Shift] = Shift;
This tells xkb to preserve the shift modifier when figuring out the appropriate level from your modifier combination in the case of LevelThree + Shift. The shift modifier is passed on for use by the toolkit or the application (see also ...

Why are they not working
According to the ArchWiki’s article that you mentioned:
X server gets keycodes from the input device and converts them to the
state and keysym.
state is the bitmask of the X modifiers (Ctrl/Shift/etc).
keysym is (according to /usr/include/X11/keysymdef.h) the integer
that
identify characters or functions associated with each ...

You can specify characters by their Unicode code point, so e.g. U0201 and U0203 should be the small a characters with the desired double grave and inverted breve. Use any Unicode table or list to find the others.
For all available key names, see /usr/include/X11/keysymdef.h. There's no name for combined character with double grave or inverted breve though.

First, you should not log in to the graphical environment as root. Could you log in as a normal user? Ubuntu doesn't support logging in as root.
Second, there is a good reason why the root account is disabled in Ubuntu. It is designed so that users that are members of the 'admin' group (e.g. the one user the installer creates) can gain root access with the ...

Before, I simply tried assigning the Pointer_Left keysym to some key, and Pointer_EnableKeys to scroll lock. However, that did not move the mouse pointer to the left. Instead, it did nothing at all.
Turns out that simply assigning a keysym to a key is not enough. The keysym needs to be interpreted as well. That is done in a "compat" file. On my system, they ...

There is a xkbvariant "altgr-intl" that puts the dead keys under the AltGr qualifier; i.e. while typing regularly, keys like " are not "dead". If you need them for international characters, you can type e.g. AltGr-" + e to get ë.
This particular layout is also available for Windows.

Solved!
The newest version of TeamSpeak3 Client is broken, so you need to install older version, because there is no Qt5 libraries for some linux distributions.
Older Client versions:
x86: http://speedy.sh/GVXsH/TeamSpeak3-Client-linux-x86-3.0.13.1.run
amd64: http://speedy.sh/vxWfy/TeamSpeak3-Client-linux-amd64-3.0.13.1.run
Remember to not update ...

Employ an extensible input method framework such as scim or ibus. The last one is possible out of the box by configuring the m17n backend, but I suspect the first two require you to program your input method.

There are a lot of bug reports about this problem. Basically, it all comes down to not logging on to Ubuntu as root, which is a huge security risk, and not advisable at all. I highly suggest you follow petersohn's advice.

Silly me! After a bit of thinking I found a good and easy way of doing it:
Copy the contents of ~/.xprofile into a script somewhere. Make sure it's executable.
Start gnome-session-properties and add a new startup program. Point it to the script.
That's it.
Much easier than fiddling around with special dot-files that are sourced on login, really :)

To get the xkb layout to show up in the gui where keyboard layouts are selected, a corresponding <variant> node has to be added to the proper <layout> section of the /usr/share/X11/xkb/rules/evdev.xml file

You can not configure it, because Alt is not a key code modifier in standard X servers. In X11 keys can be defined and changed with xmodmap. The program xev can be used to identify the key code. Start xev and press 2. On a German PC keyboard it looks this way:
KeyPress event, serial 33, synthetic NO, window 0x3000001,
root 0x69, subw 0x0, time 3044226, ...

Well, I have a silly idea: you could hack it together with xdotool and xbindkeys:
add the following to your ~/.xbindkeysrc
"xdotool key at"
Shift+Alt+Mod2 + 2
Taadaa!
(Alright, bring on the downvotes!)

The easiest way I know of is to run xev to find the keycode for your windows key (on my keyboard, it's 133, but it is likely to be different on yours), then find the keysym names for the windows key and Alt Gr (again, xev can help here, as can /usr/include/X11/keysymdef.h, which is in x11proto-core-dev on Ubuntu). To use xev, just run it from the command ...

I did this a while ago and remember I had to update
base.lst,
base.xml,
evdev.lst, and
evdev.xml,
which are all in the rules directory, on Slackware it's /etc/X11/xkb/rules.
It's possible that it would work with fewer updated files (I don't remember how much I tested), but updating these four definitely worked.